Abstract

The main topic of this thesis is the stable homotopy theory of dendroidal sets. This topic belongs to the area of mathematics called algebraic topology. Algebraic topology studies the interaction between the algebraic and topological structures.
Examples of topological spaces with a very rich algebraic structure are (iterated) loop spaces. Loop spaces carry an algebraic structure which is called an $A_{\infty}$-structure, while infinite loop spaces carry an $E_{\infty}$-structure. These structures consist of an infinite sequence of operations that satisfy various coherence laws. As it is difficult to grasp all these data, one usually uses topological operads to efficiently describe this information. One can think of operads as carrying “blueprints” for the algebraic structure which is realized in every space with that structure. The characterization results for (iterated) loop spaces using topological operads have been established in the early 1970’s by the work of P. May, M. Boardman and R. Vogt. In the 1990’s it became evident that it is important to understand the homotopy theory operads.
The theory of dendroidal sets provides a new context for studying operads up to homotopy. Dendroidal sets were introduced in 2007 by I. Moerdijk and I. Weiss. Subsequent work of I. Moerdijk and D.-C. Cisinski shows that dendroidal sets indeed model topological/simplicial operads. An important advantage of dendroidal sets is that the theory is built in a natural way as a generalization of the theory of simplicial sets. The study of dendroidal sets is very combinatorial in its nature since it is based on the notion of trees (graphs with no loops). Also, as a category of presheaves, the category of dendroidal sets has nice categorical properties.
Simplicial sets provide combinatorial models for spaces (think of it in terms of triangulations of spaces given by simplicial approximations) and dendroidal sets provide combinatorial models for infinite loops spaces as spaces together with complicated algebraic structure. In fact, the precise formulation of this idea is one of the main topics of this thesis.
A precise formulation of our results is given in the language of Quillen’s model categories. Model categories provide a formalism to study and compare homotopy theories in various contexts (topological spaces, chain complexes, simplicial sets, operads etc.) One of the main results of this thesis is that the category of dendroidal sets admits a model structure such that the underlying homotopy theory is equivalent to the homotopy theory of infinite loop spaces (equivalently, of grouplike $E_{\infty}$-algebras or connective spectra). We call this model structure the stable model structure on dendroidal sets.
Constructing a model structure is a tedious job. In our case it requires a great deal of technical combinatorial results about dendroidal sets (i.e. ab out trees). In order to simplify our arguments, in Chapter 4 we develop a combinatorial technique for proving results about dendroidal anodyne extensions. This technique can be viewed as a result in its own right as one might apply it also in different ways than it is used in the later chapters of the thesis.
We give two constructions of the stable model model structure. The first construction is more elementary and has an advantage of providing a characterization of fibrations between fibrant objects. This construction is based on standard mo del-theoretical arguments and it is given in Chapter 5.
The second construction, given in Chapter 6, is based on the work of G. Heuts. This approach makes it possible to show that the stable model structure on dendroidal sets is Quillen equivalent to a model structure on $E_{\infty}$-spaces with grouplike $E_{\infty}$-spaces as fibrant objects. The equivalence to grouplike $E_{\infty}$-objects (i.e. connective spectra) might be considered as a solution to the problem of geometric realization of dendroidal sets.
Also, these results open new possibilities to investigate the connective part of classical stable homotopy theory.
The results of the thesis presented in Chapter 7 go in that direction. In that final chapter we discuss homology groups of dendroidal sets. This homology theory generalizes the well-known homology theory of simplicial sets (i.e. the singular homology of spaces).
The generalization is not straightforward because we work with non-planar trees, but we want to use a certain sign-convention for planar trees. After giving the definition, we establish that these homology groups are homotopy invariant and that they compute the standard homology of the corresponding connective spectrum. The results of Chapters 6 and 7 are joint work with T. Nikolaus.